Approaching Throughput Optimality With Limited Feedback in Multichannel Wireless Downlink Networks

Transcription

1 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 21, NO. 6, DECEMBER Approaching Throughput Optimality With Limited Feedback in Multichannel Wireless Downlink Networks Ming Ouyang, Member, IEEE, andleiying, Member, IEEE Abstract This paper studies the allocation of feedback resources in the downlink of a frequency-division duplex (FDD) multichannel wireless system. We consider a downlink network with a single base station, shared channels, and mobile users. Throughput optimal algorithms like MaxWeight in general require the complete channel-state information (CSI) ( link states) for scheduling. Acquiring the complete CSI, however, is a prohibitive overhead in multichannel networks when the number of users is large. In this paper, we consider the scenario where the base station allocates only a limited amount of uplink resources for acquiring channel-state information. We first show that to support a fraction of the full throughput region (the throughput region with the complete CSI), the base station needs to acquire at least link states at each time-slot. We then propose a Weight-Based Feedback allocation, named WBF, and show that WBF together with MaxWeight scheduling achieves a fraction of the full throughput region by acquiring link states per time-slot. For i.i.d. ON OFF channels, we further prove that link states per time-slot is necessary for achieving a fraction of the full throughput region. Index Terms Limited feedback, MaxWeight, multichannel downlink, throughput optimality, wireless scheduling. I. INTRODUCTION T HIS paper considers wireless downlink networks using multicarrier techniques, e.g., orthogonal frequency division multiplexing (OFDM). Multicarrier techniques divide wireless resources into parallel channels, each channel corresponding to a spectrum block. To exploit multiuser diversity in multichannel downlink networks, the base station needs to acquire the channel-state information (CSI) from all mobiles Manuscript received June 09, 2012; revised November 28, 2012; accepted December 01, 2012; approved by IEEE/ACM TRANSACTIONS ON NETWORKING Editor D. Starobinski. Date of publication January 09, 2013; date of current version December 13, This work was supported by the NSF under Grants , , and and the DTRA under Grants HDTRA and HDTRA An earlier version of this work appeared in the Proceedings of the Allerton Conference on Communication, Control, and Computing 2009 and the ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc) M. Ouyang was with the Department of Electrical and Computer Engineering, Iowa State University, Ames, IA USA. He is now with The MathWorks, Inc., Natick, MA USA ( ming.ouyang@mathworks.com). L. Ying was with the Department of Electrical and Computer Engineering, Iowa State University, Ames, IA USA. He is now with the School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ USA ( lei.ying.2@asu.edu). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TNET for throughput-optimal scheduling [1], [2]. The mobiles CSI is usually unknown at the base station, especially in popular frequency-division duplex (FDD) systems that lack channel reciprocity. A common method to acquire the downlink CSI is to allocate a part of uplink resources to mobiles so that they can report their CSI to the base station. However, collecting the complete CSI is becoming increasingly expensive. Consider a multichannel downlink network with 10-MHz uplink and downlink bandwidths, respectively, divided into 50 channels (or called resource blocks). Assume there are 50 mobiles in the network. This network may have an uplink peak rate of 48 kb per subframe (1 ms) and a downlink peak rate of 80 kb per subframe [3]. In this paper, we distinguish the concepts of channel and link. A channel refers to a certain frequency band, and a link refers to the wireless connection between a mobile and the base station over a specific frequency band. Therefore, the downlink network we introduced has 50 channels and 2500 links. Assume each link state is represented by 4 bits. Then, obtaining the full CSI requires kb per subframe, which is more than 20% of the uplink capacity and 12.5% of the downlink capacity. 1 We note that for slow-fading scenarios (e.g., indoor fixed-location transmissions) where channel coherence time is large, channel states can be reported at a slower timescale (at the timescale of channel coherence time). Nevertheless, the amount of feedback resources required to collect the full CSI still linearly increases in terms of the number of channels as well as the number of mobiles, which can result in a significant communication overhead. Sincea10-MHz spectrum may cost billions of dollars (Verizon paid $4.74 billion for a block of 22-MHz bandwidth in 2008), acquiring the full CSI is clearly prohibitive in practical systems. In current and next-generation cellular standards, such as e [4], m [5], and 3GPP LTE [3], the system only allocates limited bandwidth for CSI feedback, and mobile users need to share the limited bandwidth. In this paper, we consider a multichannel downlink network with channels and mobiles. The system is operated in a slotted fashion such that the base station makes feedback/scheduling decisions once every time-slot. The duration of a time-slot can be one subframe or multiple subframes (e.g., channel coherence time). We denote the amount of the feedback resources, which is the number of link states that are allowed to be reported to the base station in one time-slot. For example, to acquire the full CSI, we need to have. 1 The control information like CSI feedback is usually transmitted at a reliable base rate, which costs more bandwidth than a regular data transmission for the same amount of information bits IEEE

2 1828 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 21, NO. 6, DECEMBER 2013 We denote the network throughput region by,whichis the set of supportable traffic loads 2 with the full CSI. In this paper, we address the following fundamental questions. 1) What is the minimum amount of feedback resources required to support a fraction of the full throughput region? 2) What is the optimal way to allocate the limited feedback resources among the mobile users? Note that the first question deals with the fundamental limit of a multichannel downlink network, and the second question deals with the design of an efficient feedback algorithm. A. Main Contributions The main contributions of this paper include the following. 1) We first establish an algorithm-independent lower bound on the amount of feedback resources required to achieve a fraction of the throughput region. We show that to achieve a fraction of the full network throughput region, the base station needs to acquire at least link states per time-slot. 2) We then develop a Weight-Based Feedback algorithm, named WBF, where the base station allocates the feedback resources among the mobiles at the beginning of each time-slot according to the queue lengths and channel statistics. We prove that WBF combined with MaxWeight scheduling achieves a fraction of the full throughput region with.wecomment that the amount of feedback resources required under WBF is a function of and and is independent of the number of mobiles in the network. The dowlink/uplink capacities of an -channel system scale linearly with, so WBF requires only a constant communication overhead. The full CSI feedback on the other hand is not scalable because it needs to acquire link states per time-slot. Besides that, we show that link states per time-slot is necessary for achieving of the full throughput region for i.i.d. ON OFF channels. We further comment that WBF is an interesting combination of a centralized resource allocation and an opportunistic feedback report. Under WBF, at the beginning of each time-slot, the base station allocates the feedback resources to mobiles (i.e., the number of link states a mobile can report), which is a centralized allocation, and then the mobiles select their best links to report, which is an opportunistic feedback report. The rest of this paper is organized as follows. First, we review related work in the rest of this section. In Section II, we introduce our system model and notation. In Section III, we show that to achieve a given fraction of the full throughput region, the amount of feedback resources required should be at least at the order of, which is independent of channel-state distributions and scheduling algorithms. In Section IV, we propose a Weight-Based Feedback scheme, which, along with the MaxWeight scheduling algorithm, achieves a fraction of the full throughput region by acquiring link states per time-slot. In Section V, we consider i.i.d. ON OFF channels and prove that link states per time-slot is also 2 Atraffic load is said to be supportable if there exists a scheduling algorithm under which the amount of backlogs at the base station is bounded. necessary in this special case. Simulation results are presented in Section VI to validate the theoretical results under different traffic and channel settings. We finally conclude in Section VII. B. Related Work There has been a lot of recent interest in developing scheduling algorithms for wireless systems with limited feedback. Opportunistic feedback in cellular downlink networks has been studied in [6] [9], where all mobiles share a common feedback channel and contend for the feedback channel if the channel state exceeds a predefined threshold. Opportunistic feedback algorithms are designed primarily for exploiting multiuser diversity and usually assume that all users are infinitely backlogged. For single-channel networks, joint channel-probing and scheduling algorithms have been proposed and analyzed in [10] [18]. In [11] [13] and [15] [18], the focus is to consider joint channel probing and scheduling problem for a time-division duplex (TDD) system. In such a system, the downlink data transmission and CSI feedback share the same radio frequency. After sending the probing request, the base station needs to wait for the feedback over the same reverse channel before transmitting downlink data. The authors in [12] modeled the joint probing and scheduling problem as a minimum cost problem and developed an algorithm with polynomial complexity. In [15], structural properties of optimal probing policy have been characterized, and in [17], a sequential probing with one bit per user overhead is studied for an OFDMA downlink system. In [18], the authors developed a throughput-optimal algorithm with limited feedback. In [10] and [14], the authors studied the joint CSI feedback and scheduling problem for FDD systems, in which CSI feedback could be transmitted simultaneously with downlink data transmissions. The work most related to ours is [14], where the optimal feedback-scheduling scheme for a single-channel downlink is derived. This paper distinguishes itself from previous works by studying joint CSI feedback and scheduling for a multichannel, multirate downlink network. The readers may ask whether the feedback-scheduling algorithms for single-channel networks can be directly applied to a multichannel network by treating the multichannel network as single-channel networks. This approach, however, does not exploit the fact that a mobile can measure the states of links after a pilot signal is broadcast from the base station and can decide which links to report (e.g., the mobile can report its best links). This additional degree of freedom makes the design of feedback-scheduling algorithms in multichannel networks fundamentally different from that in single-channel networks. For example, consider a multichannel network with symmetric channels and mobile users. The algorithms in [14] and [18] will probe the same subset of mobiles on each channel, so a mobile reports either all its link states or none. As we will see later in this paper, such schemes are suboptimal because with a high probability, only some links of a mobile are in good states. An optimal feedback scheme should allocate only a limited amount of feedback resources to one mobile, an amount sufficient for the mobile to report its good link states. Furthermore, for each channel, the base station should collect the link states of those mobiles that are likely to be scheduled, e.g., the mobiles with large queues if MaxWeight

3 OUYANG AND YING: APPROACHING THROUGHPUT OPTIMALITY WITH LIMITED FEEDBACK 1829 such that if.weassumethelinkstatesare independently distributed across both users and channels. For a mobile user, we denote the probability that link is in state by Fig. 1. Three-channel, three-user queuing system. scheduling is used. It may not be very useful to collect the link state of a mobile if the chance it gets scheduled is small. Finally, we would like to remark that communication with limited feedback has been studied in other contexts, and similar phenomenon has been discovered. For example, [19] and [20] studied the information-theoretic capacity of multiple-input multiple-output (MIMO) broadcast channels with partial CSI and proved that a near-optimal sum-throughput can be achieved by a small amount of feedback information when the number of users in the system is large. However, we comment that the focus of this paper is on the stability region (or throughput region) of wireless downlink based on queue stability and the design of feedback and scheduling algorithms to stabilize the (packet-level) queues with limited feedback. Therefore, the problem is different from the information-theoretical capacity regions with partial CSI. Furthermore, the results in [19] and [20] are the consequences of phase transitions in collections of random vectors. Our result is established by coupling the queue-based feedback allocation and queue-based scheduling. Finally, our proposed scheme can achieve a fraction of the full throughput region, i.e., the algorithm can stabilize any arrival rate such that can be stabilized under the full feedback scheme instead of just the sum capacity. II. SYSTEM MODEL AND NOTATION We consider the downlink of an FDD cellular network with onebasestation, channels, and mobile users. Each user is associated with a downlink data flow. A separate queue is maintained for each flow at the base station. Time is slotted. We use to denote the length of the queue for mobile user at the beginning of time-slot.the flows are served by the shared channels. This wireless downlink system can be modeled as a discrete-time queueing system with queues and servers as shown in Fig. 1. We use to denote the link connecting the base station and mobile using channel. The following notation is adopted throughout this paper. Denote as the link state of channel at user at time-slot. In practice, the link state is in the form of the maximum supportable data rate over that link at time-slot. In practical systems, there are a finite number of modulation and coding schemes, so we assume each link has possible link states with rates,where is the maximum transmission rate if the link is in state. For simplicity, we assume is sorted in a descending order We assume that the link-state distributions are known at the base station. 3 We further assume that, for all,,and ; for all,,and ;and,where is allowed so that a link can be off with a certain probability. In this paper, we assume the base station allocates a limited amount of uplink resources for collecting link-state information. In each time-slot, at most link states can be reported to the base station. We assume mobile knows all link states 4 and can choose a subset of them to report to the base station. We denote the feedback decision associated with link state, i.e., We further denote station on link,i.e., if is reported by user otherwise. the scheduling decision of the base if link is scheduled at time slot otherwise. We assume one time-slot is the finest granularity for feedback and scheduling, i.e., in one time-slot, a channel can be allocated tooneandatmostoneuser.asaresult We further assume a transmission to user over channel cannot be fulfilled unless the link state is reported. This assumption can be relaxed, and we will discuss the extension in Section VII. Using to denote the service rate allocated to user at time-slot,wehave Next, we define to be the number of packets arriving at time-slot for mobile user. We assume s are stationary and bounded random variables, independent across users and time-slots, independent of link states s, and. We further assume packets arrive at the base station at the beginning of a time-slot and are served at the end of the time-slot. The evolution of queue length can be written as 3 In practical cellular systems, such as the 3GPP LTE system, a low-rate feedback channel is implemented to periodically collect channel statistics from mobile users, so it is reasonable to assume that the base station knows the link-state distribution of a mobile if the mobile has been in the network for a while. 4 This is a reasonable assumption. In practical systems, the base station first broadcasts pilot signals over all channels, and then a mobile user can estimate the link states from the received pilot signals.

4 1830 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 21, NO. 6, DECEMBER 2013 TABLE I NOTATION which is identical across the mobiles. The probability that set is not empty is At each time-slot, at most one user can be served over channel, and the service rate is. Therefore, the average service rate (over channel ) allocated to user is Since the network consists of rate a user receives is channels, the average service where.wefinally recall that is the network throughput region such that given any, the network can be stabilized under some scheduling algorithm with the complete CSI. We summarize the notation in Table I. III. ALGORITHM INDEPENDENT LOWER BOUND In this section, we study the fundamental impact of limited feedback on network throughput. We are interested in knowing how many link states the base station needs to acquire to support a fraction of the full throughput region? The following theorem answers this question by providing an algorithm independent lower bound on. Theorem 1: To support a fraction of the full throughput region, the base station needs to acquire at least Hence, the traffic load for all can be supported by the scheduling algorithm above, and the traffic load lies in the network throughput region. Now under the limited feedback scheme, a link is scheduled only if the link state is reported to the base station. Therefore, at most links are scheduled at one time, and the maximum link rate is, which implies that the sum-throughput cannot exceed. We then can obtain that if is sufficient for supporting a fraction of the full throughput region, then Since the inequality holds for any,wehave link states per time-slot. Proof: Assume that for each flow constant rate.wefirst show that, packets arrive at a for all Thus, we can conclude that to achieve a fraction of the full throughput region, the amount of feedback resources should satisfy (1) is always in the network throughput region for any. To prove this, we consider the following scheduling algorithm. At time, the base station constructs a set for each channel such that if,thenuser is selected into with probability.underthis construction, mobiles have the same probability to be selected into, and the selected mobiles have. If is not empty, then the base station randomly and uniformly selects a mobile user from and serves the user with rate. Under the scheduling algorithm above, the probability that a user is selected into set is IV. ORDER-OPTIMAL FEEDBACK ALLOCATION ALGORITHM Theorem 1 implies that to achieve a fraction of throughput region, an amount of feedback resources is necessary. In this section, we propose a weight-based feedback allocation algorithm and prove that, combined with MaxWeight scheduling, the algorithm supports a fraction of the full throughput region with,which is much smaller than the amount of resources required by the full feedback. Recall in this paper that we assume the allocation of feedback resources is determined by the base station and mobile users, then decide the set of link states they want to report. Let denote the number of link states mobile user can report

5 OUYANG AND YING: APPROACHING THROUGHPUT OPTIMALITY WITH LIMITED FEEDBACK 1831 at time,so. We note there are two challenges in deciding. 1) The allocation of feedback resources is done at the base station, who does not know. Thus, mobile may not have good channels to report. 2) The mobile users cannot cooperate with each other to choose because they only know their own link states. Thus for some channel, the base station may receive multiple feedback reports from different mobiles, and for some other channels, the base station may receive none. Because of these two issues, designing the algorithm for allocating feedback resources is an interesting and challenging problem. We next introduce a weight-based allocation algorithm, called Weight-Based Feedback (WBF). The idea is to allocate the feedback resources according to queue lengths and channel statistics so that users who are more likely to have large are allocated with more feedback resources since they are likely to be scheduled under MaxWeight scheduling. We will prove that this allocation scheme, together with MaxWeight scheduling, can achieve a fraction of the full throughput region by acquiring link states per time-slot. WBF: At the beginning of time-slot, the base-station sorts in a descending order. If there is a tie, the preference is given to the smallest.define and to be the and associated with at the th position. Thus,,or with. The base station then allocates the feedback resources to mobile users iteratively as follows. Step 0: The base station sets, for all, and. Step 1: The base station sets TABLE II EXAMPLE TO ILLUSTRATE THE FEEDBACK ALLOCATION TABLE III FEEDBACK ALLOCATION WITH DIFFERENT S for all. The base station communicates the decision to mobile user and allocates corresponding uplink resources to allow mobile to report link states. Mobile, after receiving, reports the best link states to the base station. Example: The feedback resources are allocated according to the value of. Consider a simple example with two mobile users and two channels. Both channels have two states. We assume that for all,,and so that for all and, which is the average number of links in state for user. Thus, on average, user 1 has one link with rate and one link with rate.wealsoassume for simplicity. Then, the feedback resources are allocated according to Table II. Larger s are selected first. When is selected, mobile can report link states. For example, when,,,and are selected, so mobile 1 can report two link states and mobile 2 can report one link state. The feedback allocation with different s is shown in Table III. Intuition: We know that with complete CSI, the MaxWeight algorithm schedules the set of mobile users to maximize the following weighted sum: (2) and if otherwise Now with limited feedback, we need to select to make where. In other words, the base station increases the amount of feedback resources for mobile by if there are sufficient available feedback resources. Step2:If, increase by one and go to step 1. Otherwise, the base station finalizes the feedback resource allocation such that as close as (2). We therefore prefer those links with large. On average, mobile has links with weight. Hence, we should allow user to report link states if is large enough. Next, we analyze the performance of WBF when it is used with MaxWeight scheduling. MaxWeight scheduling with limited feedback is described as follows. MaxWeight Scheduling: The base station serves mobile over channel such that

6 1832 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 21, NO. 6, DECEMBER 2013 We denote by the feedback allocation decision under WBF at time and the scheduling decision under MaxWeight scheduling at time.wethendefine we have holds for all. We next consider the value of the event such that.wefirst define occurs if We analyze the performance of WBF+MaxWeight based on the following theorem [21], which reveals the fundamental relation between the efficiency of the proposed algorithm and the value of. Theorem 2: If for some, the joint WBF feedback and MaxWeight scheduling guarantees for all, then the joint algorithm can achieve a fraction of the full network throughput region. Theorem 3: Given that at most link states can be reported at each time-slot, the joint WBF and MaxWeight scheduling can supportatleasta fraction of the full throughput region, where Note that event occurs when for all mobile. Note MWBF does not report any such that,so when occurs. Next, we calculate the probability that event note that occurs only if occurs. We for all. We know that for mobile, occurs when, which happens with probability Thus, we have (3) Proof: To avoid unnecessarily complicated notation, we assume that at time, there exists a such that (4) i.e., there is no tie in allocating the feedback resources, and mobile can report link states if is selected. We emphasize that we make this assumption to simplify notation, and our analysis holds without this assumption. We next introduce a modified WBF, named MWBF. MWBF: Mobile selects the best links and forms a set named as. Link state is reported to the base station if and only if: 1) ; and 2). We note that the difference between WBF and MWBF is that MWBF will not report link state if.it is easy to see that if,then as well. Therefore, the link states reported under MWBF are a subset of those reported by WBF. Defining where, and inequality is derived from the fact that when for all and. We note that according to assumption (4) so

7 OUYANG AND YING: APPROACHING THROUGHPUT OPTIMALITY WITH LIMITED FEEDBACK 1833 We then conclude that Furthermore (5) holds for all and,so and (6) (7) We next define event such that occurs if the following two conditions hold: 1) Now, note that 2) We note that occurs implies that occurs, so We note that occurs only if there exists some mobile such that and is not reported by MWBF, which further implies that there exists a mobile such that this mobile has more than links with rate greater than or equal to and. (Therefore, mobile does not have enough feedback resources to report all links with rate greater than or equal to.) According to the Chernoff bound [22], for,the probability that mobile has more than channels with is Furthermore, when and both occur To that end, according to (5) (7), we conclude that Note that at most mobile users are allocated with nonzero feedback resources because at most different are selected at one time-slot, so We also note that the following two inequalities hold:

8 1834 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 21, NO. 6, DECEMBER 2013 and where the second inequality holds because mobile will be scheduled if then have Now we define such that.we We further note that if is too small (e.g., ), then for any. In this case, the joint WBF and MaxWeight does not provide any performance guarantee. Remark 3: In WBF, the base station needs to communicate to mobile, which incurs an additional communication overhead. This overhead is minor because: 1),so only bits are required to represent ;and2)thebase station needs to communicates with at most mobiles at each time-slot to convey. Considering the case, the base station only needs to communicates to a constant number of mobiles, and the overall communication overhead of sending is, which is a minor communication overhead because the downlink bandwidth is. In fact, this overhead can be further reduced. For example, note that if is selected for obtaining the feedback resources, then for is selected as well. Therefore, in practical implementations, the base station may just communicate the cutoff link state to mobile, which requires only bits, and mobile then can compute locally. V. SPECIAL CASE: I.I.D. ON OFF CHANNELS Since, the theorem yields from Theorem 2. Remark 1: To demonstrate the efficiency of WBF, consider the case where, which implies that. In this case decreases exponentially in terms of and is negligible compared to any constant when is sufficiently large. Therefore, for sufficiently large,toguaranteea fraction of the full throughput region, the amount of feedback resources needs to satisfy which implies that which is order-optimal because is necessary according to Theorem 1. Remark 2: From the previous remark, we can see that can be very small when is sufficiently large. Given the amount of feedback resources, the proper choice of can be obtained by finding the one that minimizes according to (3). In other words, given, we should choose (8) In this section, we consider a special case where links are i.i.d. ON OFF links, with probability of being ON.WhenalinkisON, packets can be transmitted in one time-slot. We will show that to support a fraction of the full throughput region, is necessary in this special case under the following mild assumption: After being requested to report channels, user randomly and uniformly selects on link states to report, where is the number of on links of user at time-slot. Given and, choose such that is an integer. To understand the performance of this system, we introduce an ideal channel model: At each time-slot, with probability, all users have exactly on links, and with probability, all links are ON.Thestateof the system is assumed to be independent across time-slots. It is clear that with the same amount of feedback resources, the throughput region of the ON OFF system is (strictly) contained in that of the ideal system. Next, we first study the feedback resource requirement of the ideal system. Lemma 1: Considering the ideal channel model, the maximum sum-throughput under limited feedback is when,where Proof: The maximum sum-throughput of the ideal system with feedback budget could be written as where is the maximum sum-throughput when every user has exactly on links and is the maximum sumthroughput when all links are on. (9)

9 OUYANG AND YING: APPROACHING THROUGHPUT OPTIMALITY WITH LIMITED FEEDBACK 1835 Note that when all links are ON, the sum-throughput of the system is,i.e.,. When every user has exactly ON links, the maximum sum-throughput is the solution of the following optimization problem: is strictly larger than the original one, which contradicts the assumption that is optimal. We therefore conclude that maximizes the sum-throughput for given. As a result, or.notethat since. Therefore s.t. (10) We define.notethat and is divisible by. Next, we will prove that the optimal solution of the optimization problem s.t. (11) According to (9), the maximum sum-throughput of the ideal channel model is no more than (12) Theorem 4: To support a fraction of the full throughput region under the i.i.d. ON OFF channels, it is necessary to acquire link states at each time-slot Proof: According to Lemma 1 and the fact that the maximum sum-throughput of the original system is upper-bounded by that of the ideal system, we can conclude that given, the sum-throughput of the original system with the same is no more than satisfies: or 0, ;. Note that the optimal solution must satisfy the second condition since is an increasing function of.furthermore, since each user has on links. Now suppose is not the optimal solution, then there is another feedback allocation that results in a larger but does not satisfy the first condition, i.e., there exist at least a pair of users and (assuming )suchthat and. Assuming that,wehave It is easy to check that is an achievable sumthroughput of the original system with complete CSI. Therefore, to support a fraction of the throughput region of the original system with full feedback, needs to satisfy which implies that Choosing and,wehave which yields and This implies that by increasing by one and decreasing by one, the sum-throughput under the new feedback allocation (13)

10 1836 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 21, NO. 6, DECEMBER 2013 TABLE IV POSSIBLE CHANNEL RATES TABLE V LINK-STATE DISTRIBUTION FOR THE I.I.D. CASE TABLE VI LINK-STATE DISTRIBUTIONS OF GROUP 1 AND GROUP 2 Fig. 2. Sum queue lengths of WBF, uniform, and full feedback with symmetric arrivals in the i.i.d. case. This theorem states that is necessary for achieving a fraction of the full throughput region. Recall that the proposed joint WBF and MaxWeight guarantees a fraction of the full throughput with. Whether is necessary for general channel models remains open. VI. SIMULATIONS In this section, we use simulations to evaluate WBF and compare its performance with a typical feedback allocation scheme. A. Simulation Settings We consider a downlink system with shared channels and users. A separate queue is maintained for each user. The packets of flow arrive at queue according to a Poisson process with mean arrival rate. Each channel has five possible states (rates) as listed in Table IV. We evaluate the performance of WBF in two cases: 1) link states are i.i.d. across mobile users, named as i.i.d. case; and 2) link states are independent across users but not identical, named as heterogeneous case (link states for the same user are still assumed to be i.i.d.). For the i.i.d. case, we assume the link-state distributions are identical and as shown in Table V. For the heterogeneous case, we divide the mobile users into two groups. Users in the same group have the same link-state distribution. The distributions are shown in Table VI. We can see that on average the mobiles in Group 2 have better links than those in Group 1. We use Group 2 to represent those mobiles that are close to the base station, and Group 1 to represent those mobiles that are far from the base station. In the following simulations, we compare the proposed WBF with a simple feedback resource allocation scheme that allocates the feedback resources uniformly across users, i.e., each user can report link states. The base station always uses MaxWeight scheduling based on the reported link states. B. Performance of WBF Under Various Traffic Loads In the first set of simulations, we compare the performance of WBF and the uniform allocation. We first consider the symmetric case with homogeneous arrivals, which is typical in traditional cellular network where voice is the dominant service. Fig. 3. Sum queue lengths of WBF, uniform, and full feedback with asymmetric arrivals in the i.i.d. case. Fig. 2 shows the sum queue length versus the overall arrival rate. Due to the symmetry of both arrivals and channel distributions, WBF has a similar performance with the uniform allocation. With the emergence of new applications and devices (such as video streaming and smart phones), some users require higher data rates than others. Hence, we consider asymmetric arrivals for the i.i.d. case. We include three classes of users in this simulation: two users with, three users with, and the rest of the users with,where is a positive constant to control the overall traffic load. Fig. 3 shows the sum queue lengths as a function of traffic load. We can see that WBF performs significantly better than the uniform allocation scheme. With, the sum queue length under WBF is almost identical to that under the full CSI that needs to acquire 2500 link states. The maximum throughput of the uniform allocation scheme with is only half of the one under the full feedback. The performance gain comes from the dynamic nature of WBF, which adaptively allocates the feedback resources according to users demands and leads to a more efficient resource utilization. Fig. 4 shows the standard deviations of the sum queue lengths, which behave similarly with the sum queue lengths. The plots of the standard deviations of other settings are omitted since they are also similar to the plots of the sum queue lengths. Finally, we consider asymmetric arrivals for the heterogeneous case. Again, we can observe that in Fig. 5, WBF outperforms the uniform allocation significantly. Note that with asymmetric arrivals, the uniform policy outperforms both WBF and the full feedback policy when the traffic

11 OUYANG AND YING: APPROACHING THROUGHPUT OPTIMALITY WITH LIMITED FEEDBACK 1837 Fig. 4. Standard deviations of WBF, uniform, and full feedback with asymmetric arrivals in the i.i.d. case. Fig. 5. Sum queue lengths of WBF, uniform, and full feedback with asymmetric arrivals in the heterogeneous case. load is light. This is because the MaxWeight algorithm is not delay-optimal. A better policy is an iterative longest-queue-first policy whose objective is to reduce at each time instant instead of giving priorities to long queues (details can be found in [23] [25]). Under the uniform policy, the base station is likely to serve a larger set of queues than that under WBF or the full feedback and reduces the queues more uniformly. Therefore, the uniform policy has a better performance in the light traffic regime. C. Performance of WBF With Different User Populations Theorem 3 states that to achieve a fraction of the full throughput region, WBF needs to acquire at most link states at a time, which indicates that the amount of required feedback resources is independent of the user population. We consider cases with users and and 300. To verify this result, we explicitly compute the capacity region and choose thearrivalratetobe95%of the throughput limit. Fig. 6 shows that as the number of users increases, the network is stable without increasing the amount of feedback resources, which confirms our theoretical result. The simulations above validate our analytical results. WBF approaches the full throughput region with a small amount of feedback resources, and the required amount of feedback resources is independent of the user population. VII. CONCLUSION In this paper, we considered the allocation of feedback resources in multichannel wireless downlink networks with Fig. 6. Sum queue lengths of WBF, uniform, and full feedback with different user population with symmetric arrivals in the i.i.d. case. a single base station, shared channels, and mobile users. We first showed that to support a fraction of the full throughput region (the throughput region with full channel-state information), the base station needs to acquire at least link states at each time-slot. We then proposed a feedback allocation algorithm named WBF. WBF together with MaxWeight scheduling achieves a fraction of the full throughput region by acquiring at most link states per time-slot. We have also shown that for i.i.d. ON OFF channels, under a mild assumption, acquiring link states per time-slot is necessary to support a fraction of the full throughput region. Our simulation results demonstrated asignificant performance gain of the proposed algorithm comparedtoanexistingapproach.thekeyideabehindthewbf algorithm is the queue-based resource allocation. Since long queues are more likely to be scheduled under the MaxWeight algorithm than short ones, more feedback resources should be allocated to the mobile users with long queues to guarantee that the queue weighted sum (2) is close to the one with complete link-state information. In this paper, we assumed that a transmission over link can only be fulfilled if the link state is reported to the base station. Our results, however, can be easily extended to the case where the base station can communicate with a mobile over unreported links with a base rate. In that scenario, we first use WBF to allocate the feedback resources and collect link-state information. Then, the base station serves mobile such that over channel. It can be verified that Theorem 3 still holds given. Furthermore, allowing transmissions with a base rate can only improve the throughput of the algorithm, so the order result of Theorem 2 is also valid. Therefore, our order result can be extended to the networks where blind transmissions are allowed over unreported channels with a base rate. We would like to emphasize that in this paper, we have focused on the following optimization problem: for given. It is well known that by solving this queuelength-based algorithm every time-slot, the algorithm can properly allocate the resources to users and stabilize any trafficload

12 1838 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 21, NO. 6, DECEMBER 2013 that is within the throughput region. In this paper, we only focused on the stability of the network given a set of flows with persistent arrivals. A future research problem is to consider networks with delay-sensitive flows and develop feedback allocation algorithms that can provide good delay guarantees. Finally, in this paper, we assumed the maximum transmission rate only depends on the link state and did not consider power control. Another interesting future research problem is to take power control into account and study the amount of feedback resources required. REFERENCES [1] L. Tassiulas and A. Ephremides, Dynamic server allocation to parallel queues with randomly varying connectivity, IEEE Trans. Inf. Theory, vol. 39, no. 2, pp , Mar [2] L. Tassiulas and A. Ephremides, Stability properties of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks, IEEE Trans. Autom. Control, vol. 4, no. 12, pp , Dec [3] 3GPP, 3rd Generation Partnership Project, 3GPP TS [4] IEEE Standard for Local and Metropolitan Area Networks Part 16: Air Interface for Broadband Wireless Access Systems, IEEEStd , [5] R.Vannithamby,G.Li,H.Yin,andS.Ahmadi, ProposalforIEEE m CQI feedback framework, IEEE C802.16m-08/391, [6] X. Qin and R. Berry, Opportunistic splitting algorithms for wireless networks, in Proc. IEEE INFOCOM, Hong Kong, 2004, pp [7] T. Tang and R. W. Heath, Opportunistic feedback for downlink multiuser diversity, IEEE Commun. Lett., vol. 9, no. 10, pp , Oct [8] G. de Veciana and S. Patil, Measurement-based opportunistic feedback and scheduling for wireless systems, in Proc. Annu. Allerton Conf. Commun., Control, Comput., Monticello, IL, 2005, pp [9] J. Huang, V. Subramanian, R. Agrawal, and R. Berry, Joint scheduling and resource allocation in uplink OFDM systems for broadband wireless access networks, IEEE J. Sel. Areas Commun., vol. 27, no. 2, pp , Feb [10] M. Andrews, K. Kumaran, K. Ramanan, A. Stolyar, R. Vijayakumar, and P. Whiting, Scheduling in a queueing system with asynchronously varying service rates, Prob. Eng. Inf. Sci., vol. 18, no. 2, pp , [11] Z. Ji, Y. Yang, J. Zhou, M. Takai, and R. Bagrodia, Exploiting medium access diversity in rate adaptive wireless LANs, in Proc. ACM MobiCom, Philadelphia, PA, 2004, pp [12] S. Guha, K. Munagala, and S. Sarkar, Performance guarantees through partial information based control in multichannel wireless networks, University of Pennsylvania, Philadelphia, PA, 2006 [Online]. Available: [13] A. Sabharwal, A. Khoshnevis, and E. Knightly, Opportunistic spectral usage: Bounds and a multi-band CSMA/CA protocol, IEEE/ACM Trans. Netw., vol. 15, no. 3, pp , Jun [14] A. Gopalan, C. Caramanis, and S. Shakkottai, On wireless scheduling with partial channel-state information, in Proc. Annu. Allerton Conf. Commun., Control, Comput., Monticello, IL, 2007, pp [15] N. B. Chang and M. Liu, Optimal channel probing and transmission scheduling for opportunistic spectrum access, in Proc. ACM MobiCom, Montreal, QC, Canada, Sep. 2007, pp [16] D. Zheng, W. Ge, and J. Zhang, Distributed opportunistic scheduling for ad-hoc communications: An optimal stopping approach, in Proc. ACM MobiHoc, Montreal, QC, Canada, 2007, pp [17] J. Chen, R. Berry, and M. Honig, Limited feedback schemes for downlink OFDMA based on sub-channel groups, IEEE J. Sel. Areas Commun., vol. 26, no. 8, pp , Oct [18] P. Chaporkar, A. Proutiere, H. Asnani, and A. Karandikar, Scheduling with limited information in wireless systems, in Proc. ACM MobiCom, New Orleans, LA, 2009, pp [19] C. Swannack, E. Uysal-Biyikoglu, and G. W. Wornell, Finding nemo: Near mutually orthogonal sets and applications to MIMO broadcast scheduling, in Proc. IWCMC, Maui, HI, 2005, pp [20] M. Sharif and B. Hassibi, On the capacity of MIMO broadcast channels with partial side information, IEEE Trans. Inf. Theory, vol. 51, no. 2, pp , Feb [21] A. Eryilmaz, R. Srikant, and J. R. Perkins, Stable scheduling policies for fading wireless channels, IEEE/ACM Trans. Netw.,vol.13,no.2, pp , Apr [22] M. Mitzenmacher and E. Upfal, Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge, U.K.: Cambridge Univ. Press, [23] S. Bodas, S. Shakkottai, L. Ying, and R. Srikant, Scheduling in multi-channel wireless networks: Rate function optimality in the small-buffer regime, in Proc. SIGMETRICS/Perform., Seattle, WA, 2009, pp [24] S. Bodas, S. Shakkottai, L. Ying, and R. Srikant, Low-complexity scheduling algorithms for multi-channel downlink wireless networks, in Proc. IEEE INFOCOM, San Diego, CA, 2010, pp [25] S. Bodas, S. Shakkottai, L. Ying, and R. Srikant, Scheduling for small delay in multi-rate multi-channel wireless networks, in Proc. IEEE INFOCOM, Shanghai, China, 2011, pp Ming Ouyang (S 09 M 11) received the B.E. degree in communication engineering from Beijing University of Posts and Telecommunications, Bejing, China, in 2007, and the M.S. degree in computer engineering from Iowa State University, Ames, in Currently, he is with The MathWorks, Inc., Natick, MA, working on communication and USRP-related products of MATLAB. His current research interests include wireless networking, wireless communications, and wireless system deign with software defined radio. Lei Ying (M 08) received the B.E. degree in automation from Tsinghua University, Beijing, in 2001, and the M.S. and Ph.D. degrees in electrical engineering from the University of Illinois at Urbana Champaign in 2003 and 2007, respectively. During Fall 2007, he worked as a Postdoctoral Fellow with the University of Texas at Austin. He was an Assistant Professor with the Department of Electrical and Computer Engineering, Iowa State University, Ames, from 2008 to He was the Northrop Grumman Assistant Professor (formerly the Litton Industries Assistant Professor) with the department from 2010 to He currently is an Associate Professor with the School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe. His research interest is broadly in the area of stochastic networks, including big data and cloud computing, cyber security, P2P networks, social networks, and wireless networks. Dr. Ying is an Associate Editor of the IEEE/ACM TRANSACTIONS ON NETWORKING. He won the Young Investigator Award from the Defense Threat ReductionAgency(DTRA)in2009andtheNSFCAREERAwardin2010.